Question: You have found the following ages (in years) of 6 lions. The lions are randomly selected from the 22 lions at your local zoo: $ 13,\enspace 2,\enspace 1,\enspace 5,\enspace 2,\enspace 7$ Based on your sample, what is the average age of the lions? What is the variance? You may round your answers to the nearest tenth.
Solution: Because we only have data for a small sample of the 22 lions, we are only able to estimate the population mean and variance by finding the sample mean $({\overline{x}})$ and sample variance $({s^2})$ To find the sample mean , add up the values of all $6$ samples and divide by $6$ $ {\overline{x}} = \dfrac{\sum\limits_{i=1}^{{n}} x_i}{{n}} = \dfrac{\sum\limits_{i=1}^{{6}} x_i}{{6$ To compensate for this underestimation, rather than simply averaging the squared deviations from the mean , we total them and divide by $n - 1$ $ {s^2} = \dfrac{\sum\limits_{i=1}^{{n}} (x_i - {\overline{x}})^2}{{n - 1}} $ $ {s^2} = \dfrac{{64} + {9} + {16} + {0} + {9} + {4}} {{6 - 1}} $ $ {s^2} = \dfrac{{102}}{{5}} = {20.4\text{ years}^2} $ We can estimate that the average lion at the zoo is 5 years old. There is a variance of 20.4 years $^2$.